How to run a Google Cloud Function using Go with dependencies in private repositories?

I'm trying to adapt the Go example from this tutorial to something that uses dependencies in a private repository. Here is the Cloud Function sample code:

package helloworld

import (
    "context"
    "log"

    "github.com/kurtpeek/my-private-repo/mypackage"
)

// PubSubMessage is the payload of a Pub/Sub event.
type PubSubMessage struct {
    Data []byte `json:"data"`
}

// HelloPubSub2 consumes a Pub/Sub message.
func HelloPubSub2(ctx context.Context, m PubSubMessage) error {
    name := string(m.Data)
    if name == "" {
        name = "World"
    }
    log.Printf("Hello, %s!", name)
    log.Println(mypackage.SayHello())
    return nil
}

where SayHello() is defined in the private repo github.com/kurtpeek/my-private-repo as

package mypackage

// SayHello says hello
func SayHello() string {
    return "Hello, world!"
}

Without the invocation of mypackage.SayHello(), the Cloud Function deploys and runs as expected. I'm also able to run HelloPubSub2 in a main.go after running

git config url."git@github.com".insteadOf "https://github.com"

and adding my SSH key to the repo (cf. this Medium article).

However, if I try it with the mypackage.SayHello(), I get the following error:

could not read Username for 'https://github.com'

Here is the full terminal output:

> 
gcloud functions deploy HelloPubSub2 --runtime go113 --trigger-topic mytopic
Created .gcloudignore file. See `gcloud topic gcloudignore` for details.
WARNING: Function created with limited-access IAM policy. To enable unauthorized access consider "gcloud alpha functions add-iam-policy-binding HelloPubSub2 --member=allUsers --role=roles/cloudfunctions.invoker"
Deploying function (may take a while - up to 2 minutes)...failed.                                              
ERROR: (gcloud.functions.deploy) OperationError: code=3, message=Build failed: {"error":{"buildpackId":"google.go.functions-framework","buildpackVersion":"0.9.0","errorType":13,"canonicalCode":13,"errorId":"03a1e2f7","errorMessage":"go: github.com/kurtpeek/my-private-repo@v0.0.0-20200508055124-8eb94cb388b2: invalid version: git fetch -f origin refs/heads/*:refs/heads/* refs/tags/*:refs/tags/* in /layers/google.go.functions-framework/functions-framework/pkg/mod/cache/vcs/93c0b37d34e5cf0f9b9778b99e5968bf272c89cabd21f8c0d683212c8dd2ef89: exit status 128:\n\tfatal: could not read Username for 'https://github.com': terminal prompts disabled"},"stats":null}

What is the best way to deploy this Cloud Function? Should I run go mod vendor and then change the dependencies to reference the vendor/ directory? (This seems like a hassle to do every time, though).

Update

This documentation, https://cloud.google.com/functions/docs/writing/specifying-dependencies-go#using_private_dependencies, addresses exactly this use case. I wasn't quite able to get it to work yet, though. I've added

go.mod
go.sum

to my .gcloudignore and ran go mod vendor, thereby creating a vendor/ directory, but if I try to deploy it again, I now get

cannot find package \"github.com/kurtpeek/my-private-repo/mypackage\" in any of:\n\t/usr/local/go/src/github.com/kurtpeek/my-private-repo/mypackage (from $GOROOT)\n\t/workspace/src/github.com/kurtpeek/my-private-repo/mypackage (from $GOPATH)"

~/g/s/g/k/m/helloworld> 
gcloud functions deploy HelloPubSub2 --runtime go113 --trigger-topic mytopic
Deploying function (may take a while - up to 2 minutes)...failed.                                            
ERROR: (gcloud.functions.deploy) OperationError: code=3, message=Build failed: {"error":{"buildpackId":"google.go.build","buildpackVersion":"0.9.0","errorType":2,"canonicalCode":2,"errorId":"6191efcd","errorMessage":"src/helloworld/helloworld.go:7:2: cannot find package \"github.com/kurtpeek/my-private-repo/mypackage\" in any of:\n\t/usr/local/go/src/github.com/kurtpeek/my-private-repo/mypackage (from $GOROOT)\n\t/workspace/src/github.com/kurtpeek/my-private-repo/mypackage (from $GOPATH)"},"stats":null}

I'm running this inside my GOPATH:

kurt@Kurts-MacBook-Pro-13 ~/g/s/g/k/my-cloud-function> go env GOPATH
/Users/kurt/go
kurt@Kurts-MacBook-Pro-13 ~/g/s/g/k/my-cloud-function> pwd
/Users/kurt/go/src/github.com/kurtpeek/my-cloud-function

Any ideas how to fix this?